In Eq. (REF ), \(k_x\)  and \(k_z\)  are the real wave numbers in the \(x\)  and \(z\)  directions, respectively, and \(\omega \)  is the complex frequency.
Because \(c_x, k_x,c_z,k_z\)  are real, \(\omega \)  is always real, just as in the case of water waves [1]}, [2]}. As such, the imaginary part of \(\omega \)  is always 0, and all modes –Eq (REF )– are neutrally stable for all values of the parameters according to Eq. ().
As a check, note that the 1D dispersion relation (Equation (5a) in [3]} with \(n=2\) ) is recovered by letting \(c_z=k_z=0\)  in Eq. (REF ).
According to classical stability analysis in Eq. (), the response to such a disturbance should neither grow nor decay. However, the solution grows algebraically in 1D as shown in King et al. [3]}.
